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J. Phys. Chem. B 2000, 104, 6258-6265
Correlation between Local and Long-Range Structure in Compressible Supercritical Lennard-Jones Fluids: State-Point Dependence Grant Goodyear, Michael W. Maddox, and Susan C. Tucker* Department of Chemistry, UniVersity of California at DaVis, DaVis, California 95616 ReceiVed: January 20, 2000; In Final Form: April 17, 2000
Evaluation of local-density-specific radial distribution functions in a two-dimensional supercritical LennardJones fluid has enabled us to elucidate how the correlations between a solvent particle’s local density and the extended surrounding solvent structure change as a function of the thermodynamic condition. We find that the standard correlation length provides only a very crude measure of these correlations and that, for example, such inhomogeneity-induced correlations remain present at bulk densities surprisingly far from the critical density in this two-dimensional system. Additionally, these local-density-specific radial distribution functions provide insight into the state-point dependence of the origin of mean local-density enhancements, showing that the potential-induced and the correlation-induced contributions both die out rapidly with increasing temperature, whereas they are maximally important at different bulk densities.
I. Introduction The compressible supercritical regime,1,2 which spans a broad region of the phase diagram surrounding the critical point, may potentially be accessed during processing in supercritical fluids,1,3-12 either intentionally or accidentally, due to nonuniform conditions and/or heat-up and cool-down phases. It would therefore be beneficial to process design if we had the ability to predict chemical reactivity in this regime,1,4,6,12 Since in many chemical systems solute dynamics and reactivity are controlled by the local solvent environment around the solute, it is the local, rather than the global, properties of the solvent which we must understand in order to gain such predictive abilities. Thus, we ask what is the nature of the local solvent environments supported by these highly compressible fluids? And what, if any, is the relationship between these local environments and the extended density correlations13-16 which characterize these fluids (Figure 1)? There has been much work to date, both experimental and theoretical, toward understanding the mean local solvent environment around solutes in compressible supercritical fluids.1,12 This work has demonstrated that, in the compressible regime, many solutessincluding any tagged solvent molecule in a neat supercritical fluidswill experience a mean local density which is enhanced relative to the bulk value. Some solutes, on the other hand, experience a lowerthan-bulk mean local density, and it is known that whether a solute will have an enhanced or depleted local solvent density will depend on the strength of the solute-solvent interaction potential (relative to that of the solvent-solvent interaction potential).13,17-19 However, focus has traditionally been confined to determination of this aVerage local density around a solute or tagged solvent molecule, and it is only very recently that the distributions of local densities found in these fluids have been examined.2,13,20,21 Perhaps surprisingly, examination of the distribution of local densities found around solvent atoms in neat supercritical fluids, P (Fl ), has shed light on the relationship between the local and long-range density fluctuations.2,13,15 Indeed, in an earlier work2 we showed how the presence of critical, extended density inhomogeneities will cause the distribution, P (Fl ), of this atom-
centered function, Fl , to shift toward higher densities, causing a concomitant enhancement of the mean local density, 〈Fl 〉. Yet, as we discussed in that work,2 these extended fluctuations are not the only source of density inhomogeneities in a real supercritical fluid; there also exist short-range, attractive potential interactions which generate small, but relatively longlived, “clusters” of atoms at low bulk densities. If the local region being considered is sufficiently small, these clusters will themselves generate mean local-density enhancements. Hence, there are two contributions to the local-density enhancements: “correlation-induced”, which arise from the long-length-scale critical fluctuations and are maximized at the critical point, and “potential-induced”, which arise from the short-range potential interactions and tend to be maximized at rather low densities. In order to further clarify the relationship between the local densities and these two inhomogeneity mechanisms, we recently introduced a structural correlation function, g(r|Fl ), which gives the probability of finding an atom a distance r away from a central atom giVen that this central atom is surrounded by a local density of Fl .15 Examination of this correlation function for a neat two-dimensional supercritical fluid Lennard-Jones fluid at the compressible state point shown in Figure 1a demonstrated a clear correlation between specific local densities and the surrounding long-range fluid structure. Additionally, we saw that a detailed analysis of this correlation could provide insight into the source of the local-density fluctuations observed. Hence, in the present work, we make use of this new function to examine how the nature of the local-density fluctuations change with both temperature (T) and bulk density (F), paying particular attention to the extent to which the correlation-induced and potential-induced mechanisms contribute to the observed mean local-density enhancements. In section II we briefly introduce the set of local-densityspecific distribution functions {g(r|Fl )}. In section III we briefly describe the simulation details, and section IV contains the results. In particular, in section IVA we illustrate the interpretation of the functions {g(r|Fl )} via application to a single supercritical state point, while in section IVB we use this understanding to investigate the nature of the local and long-
10.1021/jp000380a CCC: $19.00 © 2000 American Chemical Society Published on Web 06/10/2000
Structure in Compressible Supercritical LJ Fluids
J. Phys. Chem. B, Vol. 104, No. 26, 2000 6259 distance r away from the origin, given that there is a particle at the origin surrounded by a local density of Fl ; the fact that a fluid supports a distribution of local densities requires that we analyze the set of local-density-specific radial distribution functions, {g(r|Fl )}, corresponding to that distribution. For ease of interpretation, g(r|Fl ) is actually this conditional probability normalized by the equivalent conditional probability for an ideal gas.15 In this way, purely statistical fluctuation effects are removed from the functions {g(r|Fl )}. The instantaneous, atom-centered local density, Fl (t), used to define these local-density-specific functions is simply the number of atoms found within a radius rl of the specified central atom at time t, divided by the appropriate local volume. See ref 2 for details. The local-density-specific radial distribution functions {g(r|Fl )} were derived previously,15 so here we present only the expressions used to evaluate these functions from the molecular dynamics simulations. Specifically
g(r|Fl ) ) lim f0
〈N(r|Fl )〉
(2)
〈Nideal (r|Fl )〉
where 〈N (r|Fl )〉 is the average number of solvent particles found within a spherical shell of width centered a distance r away from a tagged solvent particle, giVen that the tagged particle has a surrounding local density of Fl . Because the density inside the local radius is constrained to be Fl , the ideal gas denominator of eq 2 is (for sufficiently large systems, see ref 15)
1 2 1 2 〈Nideal (r|Fl )〉 ) Fl π r + - π r - , r < rl 2 2
[ ( ) ( )] 1 1 ) F[π(r + ) - π(r - ) ], 2 2 2
2
r > rl (3)
where F is the bulk density and we have assumed a twodimensional system. In comparison, the usual radial distribution function, g(r), which can be obtained from a properly weighted average of {g(r|Fl )} over Fl ,15 is given by22
g(r) ) lim
f0+
〈N(r)〉
(4)
〈Nideal (r)〉
Here, 〈N (r)〉 is the average number of solvent particles found within a spherical shell of width centered a distance r from a tagged solvent particle, and the corresponding number expected for an ideal gas is, in a two-dimensional system,
1 2 1 2 (r)〉 ) F π r + - π r - 〈Nideal 2 2
[(
Figure 1. Representative snapshots of a neat supercritical LennardJones fluid in two dimensions and at different thermodynamic state points: (a) T ) 0.55 /kB, F ) 0.3 σ-2; (b) T ) 5.0 /kB, F ) 0.3 σ-2; (c) T ) 0.55 /kB, F ) 0.1 σ-2.
)
(
)]
(5)
Finally, we note that the correlation length, ξ, which is frequently quoted as a measure of the length scale of the density inhomogeneities in a compressible fluid, is defined as the exponential decay constant of the total g(r) at large r (in d dimensions):23,24
[g(r) - 1] ∼ r-(d-1)/2e-r/ξ
(6)
range density inhomogeneities as a function of the thermodynamic conditions. Conclusions follow in section V.
III. System and Simulation Details
II. Local-Density-Specific Radial Distribution Functions The local-density-specific radial distribution function, g(r|Fl ), gives the conditional probability of finding a solvent particle a
All details about the system and the simulation may be found in ref 25. Note that we paid very careful attention to the convergence of these simulations, and specifically, to the
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Figure 2. Temperature dependence of the partial radial distribution functions (rl ) 3.09 σ, rexcl ) 0.78 σ). The percentage listed in each panel gives the percentage of particles having the given local density. (a, left column) T ) 0.55 /kB, F ) 0.3 σ-2. (b, middle column) T ) 0.85 /kB, F ) 0.3 σ-2. (c, right column) T ) 5.0 /kB, F ) 0.3 σ-2. The bottom panel of each column is the total radial distribution function, g(r), at that state point.
convergence of the distribution functions g(r|Fl ) with respect to system size and overall simulation time, as discussed in ref 15. IV. Results A. Interpretation of g(r|Fl ). In Figure 2a we show five of the functions {g(r|Fl )} for the compressible state point shown in Figure 1a (T ) 0.55 /kB, F ) 0.30 σ-2).15 There are, in fact, 23 g(r|Fl ) functions at this state point, corresponding, via Fl ) Nl /Vl , to each of the integer numbers of particles, Nl , found to fall within the local region. Figure 2a gives only a representative sample of these functions, including gaslike Fl ) 0.07 σ-2, intermediate Fl ) 0.28 and 0.38 σ-2, liquidlike, Fl ) 0.61 σ-2, and solidlike, Fl ) 0.75 σ-2, local densities. We remind the
reader that the short-range, long-range, and discontinuity behaviors of these functions provide information about the mechanistic origins of the mean local-density enhancements, 〈Fl 〉 > F,2 observed in compressible supercritical fluids (Table 2 of ref 2), as well as about the presence of any distribution of extended structures (that is, structures extending beyond the range of the local region).15 The short-range part of these local-density-specific radial distribution functions, g(r F.2 The qualitative importance of such “potential-induced” contributions to the local density enhancement can be deduced by examining the strength and sharpness of the nearest-neighbor peak in the short-range g(r|Fl ). In Figure 2a the very distinct first peak observed in g(r|Fl ) for all of the local densities, including the very lowest, indicates that such potential-induced clusters are indeed formed at this temperature and will play a role in local density enhancement formation, especially at low bulk densities, where the presence of such clusters generates a significant enhancement of Fl over F. (Note that structural contributions to the local density enhancement, which are expected at high densities, have been largely eliminated by our choice of rl and rexcl in the definition of Fl ; see ref 2). On the other hand, the presence and decay of a long-range tail in g(r>rl |Fl ), which is normalized by the bulk density, eq 3, is indicative of a correlated-density contribution to the local density enhancement. Indeed, in Figure 2a, one observes a strong correlation between the local density and the sign of the decay of the long-range tail, with low (high) local densities exhibiting a decay from below (above). This local-to-long-range density correlation is a reflection of the fact2,15 that low (high) local densities tend to be found embedded well within extended low (high) density domains and thus reflect the correlation of the density from its low (high) value inside the local region (r < rl ) to a similarly low (high) value outside the local region (r > rl ). Consequently, such correlated-density effects arise only in fluids having long-length-scale density inhomogeneities (e.g., those due to critical fluctuations) and thus they serve to signal the presence of such inhomogeneities. Moreover, these density inhomogeneities themselves contribute to the mean local density enhancementsthe “correlation-induced” contributionssimply because more solvent atoms are found in the high-density domains than in the low, such that Fl > F is sampled with greater frequency than is Fl < F. Note that this nonuniform sampling explains why the tails of opposing sign in {g(r|Fl )} do not entirely cancel in the averaging to yield g(r)sthe functions g(r|Fl >F) are weighted more heavily than are g(r|Fl rl will cause a lowdensity local environment and a step-up discontinuity as r crosses rl , while a fluctuation from r > rl to r < rl will yield a high-density local environment and a step-down discontinuity. The correlated-density discontinuity, on the other hand, arises because the discontinuity which appears in the ideal gas normalization of g(r|Fl ) (eq 3) is, in inhomogeneous fluids, reduced by the density correlation across the rl boundary, and, as a result, this discontinuity is not completely canceled in g(r|Fl ) at rl .15 Thus, this source of discontinuities arises from the same correlations as do the extended long-range tails in g(r|Fl ), from which it follows that the correlated-density contribution to the discontinuity at rl will (i) appear only in inhomogeneous fluids, and (ii) have a ( “sign” which corre-
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Figure 3. Discontinuities in the partial radial distribution functions d(Fl ) as a function of the local density, using rl ) 3.09 σ and rexcl ) 0.78 σ. (a, top) Temperature dependence at F ) 0.3 σ-2. (b, bottom) Density dependence at T ) 0.55 /kB. To ensure that the data plotted is statistically meaningful, we have plotted only those values of Fl for which the percentage of particles having that Fl exceeds 0.05% (one particle in 2000) under the indicated thermodynamic conditions.
sponds to the nature of the long-range decay, with a step-down (step-up) discontinuity corresponding to a long-range decay from below (above) as expected around a low (high) local-density environment. In ref 15, it was shown that the structural effect, by itself, goes monotonically from step-up to step-down as Fl increases, while the correlated-density effect, by itself, behaves oppositely, going monotonically from step-down to step-up. (Note that as these two effects generally do not cross zero at the same local density, any observed disappearances of the total discontinuity will most likely represent a cancellation of the two contributions.) Consequently, if one examines the discontinuities vs Fl in a real fluid, one will observe step-up to step-down behavior only if the fluid is homogeneous, and step-down to step-up (possibly returning to step-down at high local densities) when inhomogeneities are present.15 To present the local-density dependence graphically, we assign a value to each discontinuity according to d(Fl ) ≡ g(r+ l | Fl ) - g(rl | Fl )
(7)
where r+ l and rl are positions infinitesimally greater and less than rl , respectively, and d(Fl ) > 0 and d(Fl ) < 0 correspond to step-up and step-down discontinuities, respectively. The solid line in Figure 3a shows d(Fl ) vs Fl for the g(r|Fl ) of Figure 2a. The step-down, step-up, step-down behavior confirms the existence of the inhomogeneities extending over lengths greater than 2rl which are observed in the snapshot Figure 1a. We emphasize that these discontinuities provide a more informative indicator of the extent of the long-range density inhomogeneities than does the usual correlation length ξ ) 3.3 σ or 2ξ ) 6.6 σ,26 at this state point. As just discussed, the solid line in Figure 3a confirms that density inhomogeneities
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Figure 4. Bulk-density dependence of the partial radial distribution functions at T ) 0.55 /kB, with rl ) 3.09 σ and rexcl ) 0.78 σ. The various columns correspond to different bulk densities. The percentage given in each panel is the percentage of particles having the corresponding local density.
of lengths greater than 2rl ) 6.2 σ ∼ 2ξ, i.e., greater than the correlation length, are present. In fact, a d(Fl ) plot based on rl ) 4.72 σ (not shown) also exhibits this characteristic pattern of inhomogeneous fluids, indicating that the density inhomogeneities at this state point exceed even 9.44 σ. Additionally, the lengths predicted by d(Fl ) are more consistent with the observed extent of the inhomogeneities than is 2ξ. Specifically, the local-to-long-range correlations are observed (Figure 2a) to extend, on average, out to r ∼ 9 σ, such that 2r ∼ 18 σ . 2ξ ) 6.6 σ. Similarly, the inhomogeneities themselves in the snapshot shown in Figure 1a are observed to extend over ranges much greater than 2ξ ) 6.6 σ. B. State-Point Dependence of g(r|Gl ). Short-range Structure and the “Potential-induced” Effect. Looking first at the effect of temperature on g(r
